Lame duck Los Angeles Mayor Antonio Villaraigosa must be feeling like the last man standing in a game of musical chairs. Having campaigned tirelessly for President Obama’s re-election, and having chaired last summer’s successful Democratic National Convention, Villaraigosa was expected to join the administration in Washington, reports The Post’s Richard Johnson. Instead, he’s gotten the cold shoulder. “They’ve shown him no respect, or courtesy,” said an insider. Obama lost two Latino Cabinet members — Ken Salazar and Hilda Solis — from his first term, and so far has none in his new Cabinet. “But they have Villaraigosa twisting in the wind,” said our source. “It’s gotten very testy.”